Complete corrected formula for generating functions of the hypergeometric distribution

The hypergeometric distribution is a popular distribution, whose properties have been extensively investigated. Generating functions of this distribution, such as the probability-generating function, the moment-generating function, and the characteristic function, are known to involve the Gauss hypergeometric function. We elucidate that the existing formula of generating functions is incomplete and provide the corrected formula. In view of the utility and applicability of the hypergeometric distribution, the exact and correct foundation is crucially important.